Harmonic operators: the dual perspective

The study of harmonic functions on a locally compact group G has recently been transferred to a “non-commutative” setting in two different directions: C.-H. Chu and A. T.-M. Lau replaced the algebra L∞(G) by the group von Neumann algebra VN(G) and the convolution action of a probability measure μ on L∞(G) by the canonical action of a positive definite function σ on VN(G); on the other hand, W. Jaworski and the first-named author replaced L∞(G) by B(L(G)) to which the convolution action by μ can be extended in a natural way. We establish a link between both approaches. The action of σ on VN(G) can be extended to B(L(G)). We study the corresponding space H̃σ of “σ-harmonic operators”, i.e., fixed points in B(L (G)) under the action of σ. We show, under mild conditions on either σ or G, that H̃σ is in fact a von Neumann subalgebra of B(L(G)). Our investigation of H̃σ relies, in particular, on a notion of support for an arbitrary operator in B(L(G)) that extends Eymard’s definition for elements of VN(G). Finally, we present an approach to H̃σ via ideals in T (L(G))—where T (L(G)) denotes the trace class operators on L(G), but equipped with a product different from composition—, as it was pioneered for harmonic functions by G. A. Willis.